# 布巴克尔多項式

## 布巴克尔多項式

{\displaystyle {\begin{aligned}B_{0}(x)&{}=1\\B_{1}(x)&{}=x\\B_{2}(x)&{}=x^{2}+2\\B_{3}(x)&{}=x^{3}+x\\B_{4}(x)&{}=x^{4}-2\\B_{5}(x)&{}=x^{5}-x^{3}-3x\\B_{6}(x)&{}=x^{6}-2x^{4}-3x^{2}+2\\B_{7}(x)&{}=x^{7}-3x^{5}-2x^{3}+5x\\B_{8}(x)&{}=x^{8}-4x^{6}+8x^{2}-2\\B_{9}(x)&{}=x^{9}-5x^{7}+3x^{5}+10x^{3}-7x\\&{}\,\,\,\vdots \end{aligned}}}

{\displaystyle {\begin{aligned}B_{0}(x)&=1,\\B_{1}(x)&=x,\\B_{2}(x)&=x^{2}+2,\\\end{aligned}}}

{\displaystyle {\begin{aligned}B_{m}(x)&=xB_{m-1}(x)-B_{m-2}(x)\quad {\text{, }}m>2.\end{aligned}}}

${\displaystyle \sum _{n=0}^{\infty }{\tilde {B}}_{n}(x)t^{n}={\frac {1+3t^{2}}{1-t(t-x)}}.\,\!}$

### 生成解

${\displaystyle B_{n}(x)=\sum _{p=0}^{\lfloor n/2\rfloor }{\frac {n-4p}{n-p}}{\binom {n-p}{p}}(-1)^{p}x^{n-2p}}$

### 微分操作代表

${\displaystyle (x^{2}-1)(3nx^{2}+n-2)y{''}+3x(nx^{2}+3n-2)y{'}-n(3n^{2}x^{2}+n^{2}-6n+8)y=0\,}$

## 参考文献

1. ^ Oyodum, OD; Awojoyogbe, OB; Dada, MK.; Magnuson, JN, On the earliest definition of the 布巴克尔 polynomials, Eur. Phys. J. Appl. Phys.: 21201–21202[永久失效連結]
2. ^ Sequences A135929 and A135936 by Neil J. A. Sloane, A137276 , Roger L. Bagula , Gary Adamson,A138476, A. Bannour, A137289, A136256, A136255 , R. L. Bagula 在 On-Line Encyclopedia of Integer Sequences
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5. ^ A. Milgeam, The stability of the 布巴克尔 polynomials expansion scheme (BPES)-based solution to Lotka–Volterra problem, Journal of Theoretical Biology (Elsevier), {{doi:10.1016/j.jtbi.2010.01.026}}
6. ^ H. Koçak, A. Yıldırım, D.H. Zhang, S.T. Mohyud-Din, The Comparative 布巴克尔 Polynomials Expansion Scheme (BPES) and Homotopy Perturbation Method (HPM) for solving a standard nonlinear second-order boundary value problem, Mathematical and Computer Modelling(Elsevier), {{doi:10.1016/j.mcm.2011.02.031 }}
7. ^ A. Yildirim, The 布巴克尔 polynomials expansion scheme for solving nonlinear science problems, (PDF), The 7th International Conference on Differential Equations and Dynamic Systems, University of South Florida, Tampa, Fmorida USA, 15-18 December 2010 <Page 40 >, （原始内容 (PDF)存档于2011年7月22日）
8. ^ Paul Barry, Aoife Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, Chapter 6: The 布巴克尔 polynomials (PDF), Journal of Integer Sequences (JIS)
9. ^ H. Koçak, Z. Dahong, A. Yildirim, A range-free method to determine antoine vapor-pressure heat transfer-related equation coefficients using the 布巴克尔 polynomials expansion scheme, Russian Journal of Physical Chemistry A, Focus on Chemistry (Springer)[永久失效連結]
10. ^ H. Koçak, Z. Dahong, A. Yildirim, Analytical expression to temperature-dependent Kirkwood-Fröhlich dipole orientation parameter using the 布巴克尔 Polynomials Expansion Scheme (BPES), Indian Journal of Physics(Springer)[永久失效連結]
11. ^ A. Belhadj, O. F. Onyango and N. Rozibaeva, 布巴克尔 Polynomials Expansion Scheme-Related Heat Transfer Investigation Inside Keyhole Model (PDF), Jornal of Thermophysics and Heat Transfer (American Institute of Aeronautics and Astronautics) AIAA)[永久失效連結]
12. ^ D. H. Zhang, Study of a non-linear mechanical system using 布巴克尔 polynomials expansion scheme BPES, International Journal of Non-Linear Mechanics(Elsevier)
13. ^ Emna Gargouri-Ellouze, Noreen Sher Akbar, Sohail Nadeem, Modelling Nonlinear Bivariate Dependence Using the 布巴克尔 Polynomials Copula The 布巴克尔 polynomials (PDF), Studies in Nonlinear Sciences (SNS)
14. ^ W. X. Yue, H. Koçak, D. H. Zhang , A. Yıldırım, A second attempt to establish an analytical expression to steam-water dipole orientation parameter using the 布巴克尔 polynomials expansion scheme, Journal of Structural Chemistry (Springer)[永久失效連結]
15. ^ D. H. Zhang, L. Naing, The 布巴克尔 polynomials expansion scheme BPES for solving a standard boundary value problem (PDF), Applied Sciences,(Balkan Society of Geometers, Geometry Balkan Press)
16. ^ A. Belhadj, J. Bessrour, M. Bouhafs and L. Barrallier, Experimental and theoretical cooling velocity profile inside laser welded metals using keyhole approximation and 布巴克尔 polynomials expansion, Journal of Thermal Analysis and Calorimetry(Akadémiai Kiadó, Springer Science & Kluwer Academic Publishers B.V.), {{doi:10.1007/s10973-009-0094-4 }}[永久失效連結]
17. ^ S. Amir Hossein A. E. Tabatabaei, T. Gang Z., O. Bamidele A. and Folorunsho O. Moses, Cut-off cooling velocity profiling inside a keyhole model using the 布巴克尔 polynomials expansion scheme, Heat and Mass Transfer(Springer Berlin / Heidelberg), Volume 45, Number 10 / août 2009, pages:1247-1251 {{doi:10.1007/s00231-009-0493-x}}[永久失效連結]
18. ^ S. Fridjine and M. Amlouk, A NEW PARAMETER-ABACUS FOR OTIMIZING PV-T HYBRID SOLAR DEVICES FUNCTIONAL MATERIALS USING 布巴克尔 POLYNOMIALS EXPANSION SCHEME, Modern Physics Letters B ([ISSN: 0217-9849, by WS: World Scientific Publishing Co Pte Ltd] )
19. ^ T. G. Zhao, Y. X. Wang and K. B. Ben Mahmoud, Limit and uniqueness of the 布巴克尔-Zhao polynomials single imaginary root sequence, International Journal of Mathematics and Computation, ISSN 0974-5718, （原始内容存档于2011-08-13）
20. ^
21. ^ B. Tirimula Rao, P. Srinivsu, C. Anantha Rao, K. Satya Vivek Vardhan, Jami Vidyadhari ,Page 8 : 布巴克尔 polynomials
22. ^ Kiliç Bülent, Erdal Bas, Page 7, Citation 27: 布巴克尔 polynomials

## 外部链接

a New Parameter:. AN Abacus for Optimizing Pv-T Hybrid Solar Device Functional Materials Using the 布巴克尔 多項式 Expansion Scheme and [1]
De Khawarizmi à Euler, La Presse Magazine, January 9, 2008 （法文）, vedi anche tunisie7arts.com
Le polynôme de 布巴克尔 (PDF), La Presse Magazine, April 22, 2007, (1019): 6[永久失效連結] （法文）
A new polynomial sequence... The 布巴克尔 Polynomials, Numerical Methods for Partial Differential Equations NMPDE[永久失效連結]
Solution to Heat Equation Using 布巴克尔 Polynomials (PDF), J. of Thermophysics and Heat Transfer[永久失效連結]

• WS World Scientific Publishing Co Pte Ltd
AMLOUK–布巴克尔 EXPANSIVITY..USING 布巴克尔 多項式, Functional Materials Letter